A hemisphere is mounted on top of a
cube of side 10 centimeters, with its flat face resting on the cube. If the flat face of the hemisphere
is completely contained by the adjoining face of the cube, what is the largest
diameter that the cube can have? Find the cost of painting the
surface of the combined solid, with a hemisphere of maximum diameter, at a rate of
five rupees per centimeter squared. Use 𝜋 equals 3.14.
So in this question, we have a
hemisphere mounted on top of a cube with the flat face of the hemisphere resting on
the cube. So let’s draw a quick sketch of
what this would look like. We’re also told that the cube has a
side length of 10 centimeters. So we can label this on our
Another crucial piece of
information that the question tells us is that the flat face of the hemisphere is
completely contained by the adjoining face of the cube. And what this means is that the
flat face of the hemisphere which is touching the cube is contained within the
square top of the cube. And what the question is asking is
what’s the maximum diameter of this circular face of the hemisphere such that it
remains completely contained within the face of the cube.
Now, the largest possible
hemisphere which could remain contained within the square face of the cube would
have the two widest parts of its circular face touching the edges of the square face
of the cube. Now, let’s look at a top-down view
of what this would look like.
Now, the circle in this diagram
will be identical to the circular face of the hemisphere. And as we can see, the widest part
of the circle is touching the edges of the square. And now, as we know, the widest
part of the circle is actually its diameter. And since the square is one of the
faces of our cube, we know that it has a side length of 10 centimeters. This means that the largest
diameter of this circle and therefore of the hemisphere is 10 centimeters.
So now, we have answered the first
part of this question. Next, we’re asked to find the cost
of painting the surface of this combined solid, where the hemisphere has a maximum
diameter. And this is in fact what we’ve just
found. Now, in order to find the cost of
painting this solid, we first need to find its surface area.
Now, if we’re finding the surface
area of the shape, we can look at our diagram to help us out. So the total surface area will be
equal to the surface area of the cube minus the area of the circular face of the
hemisphere, which is touching the cube, plus the surface area of the curved part of
Now, let’s calculate each of these
parts individually. In order to find the surface area
of the cube, we can find the surface area of each individual face since the cube has
six identical square faces. So the surface area of one of the
cubes’ faces is simply is length squared. So that’s 10 times 10 centimeters,
which gives us 100 centimeters squared. And since there are six square
faces on a cube, that means that the total surface area of the cube is six times 100
centimeters squared, which is equal to 600 centimeters squared.
Next, we’ll find the area of the
flat circular face of the hemisphere. Now, we know that the circle has a
maximum possible diameter. And since the radius is equal to
half of the diameter, this means that the radius must be equal to 10 over two
centimeters or five centimeters.
And now, the area of a circle is
given by the formula 𝜋𝑟 squared, where 𝑟 is the radius of the circle. And since the radius of our circle
here is five centimeters, this means that its area is 𝜋 timesed by five centimeters
squared or 25𝜋 centimeters squared.
Now that we found the radius of our
hemisphere, it’ll be easier to calculate the curved surface area of it. And now, the surface area of a
sphere is equal to four 𝜋𝑟 squared. And so the surface area of the
curved part of the hemisphere will be equal to half of this amount, which is also
equal to two 𝜋𝑟 squared.
So in order to find our curved
surface area, we simply substitute 𝑟 equals five centimeters into this
equation. And this gives us two 𝜋 times five
centimeters squared. And this is simply equal to 50𝜋
centimeters squared. So now, we have found the total
surface area, we just need to simplify this equation.
We have that this is equal to 600
plus 25𝜋 centimeters squared. And now, we can substitute in the
approximation of 𝜋 of 3.14. Multiplying the 3.14 by 25, we
obtain 78.5. And for our final step in finding
the total surface area of this solid, we simply add the 600 and the 78.5 to leave us
with 678.5 centimeters squared.
Now, all that remains to do is to
find the cost of the paint to cover this total surface area. So we have that the cost of the
paint is five rupees per centimeter squared. So the total cost is equal to the
total surface area, which is 678.5, multiplied by the rate, which is five. And since 678.5 multiplied by five
is equal to 3392.5, the total cost of painting this combined solid is 3392 rupees
and 50 paise.